# Properties

 Label 495.6.a.b Level $495$ Weight $6$ Character orbit 495.a Self dual yes Analytic conductor $79.390$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,6,Mod(1,495)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("495.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$495 = 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 495.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$79.3899908074$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.18257.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 26x + 8$$ x^3 - x^2 - 26*x + 8 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 - 14) q^{4} - 25 q^{5} + ( - 6 \beta_{2} - 20 \beta_1 - 14) q^{7} + ( - 2 \beta_{2} - 37 \beta_1 + 21) q^{8}+O(q^{10})$$ q + (b1 - 1) * q^2 + (b2 - b1 - 14) * q^4 - 25 * q^5 + (-6*b2 - 20*b1 - 14) * q^7 + (-2*b2 - 37*b1 + 21) * q^8 $$q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 - 14) q^{4} - 25 q^{5} + ( - 6 \beta_{2} - 20 \beta_1 - 14) q^{7} + ( - 2 \beta_{2} - 37 \beta_1 + 21) q^{8} + ( - 25 \beta_1 + 25) q^{10} - 121 q^{11} + (44 \beta_{2} - 24 \beta_1 + 90) q^{13} + ( - 14 \beta_{2} - 68 \beta_1 - 278) q^{14} + ( - 67 \beta_{2} + 35 \beta_1 - 186) q^{16} + (102 \beta_{2} - 236 \beta_1 - 100) q^{17} + ( - 46 \beta_{2} + 172 \beta_1 - 994) q^{19} + ( - 25 \beta_{2} + 25 \beta_1 + 350) q^{20} + ( - 121 \beta_1 + 121) q^{22} + ( - 64 \beta_{2} - 688 \beta_1 + 464) q^{23} + 625 q^{25} + ( - 68 \beta_{2} + 486 \beta_1 - 850) q^{26} + (138 \beta_{2} + 236 \beta_1 - 318) q^{28} + ( - 330 \beta_{2} - 652 \beta_1 + 1840) q^{29} + ( - 44 \beta_{2} - 56 \beta_1 - 4956) q^{31} + (166 \beta_{2} + 395 \beta_1 + 645) q^{32} + ( - 338 \beta_{2} + 818 \beta_1 - 4728) q^{34} + (150 \beta_{2} + 500 \beta_1 + 350) q^{35} + (648 \beta_{2} - 448 \beta_1 - 2130) q^{37} + (218 \beta_{2} - 1408 \beta_1 + 4286) q^{38} + (50 \beta_{2} + 925 \beta_1 - 525) q^{40} + (330 \beta_{2} + 1804 \beta_1 + 2264) q^{41} + (618 \beta_{2} + 1340 \beta_1 - 11850) q^{43} + ( - 121 \beta_{2} + 121 \beta_1 + 1694) q^{44} + ( - 624 \beta_{2} - 112 \beta_1 - 11648) q^{46} + ( - 364 \beta_{2} + 1544 \beta_1 + 7820) q^{47} + (280 \beta_{2} + 2832 \beta_1 - 5935) q^{49} + (625 \beta_1 - 625) q^{50} + ( - 854 \beta_{2} - 694 \beta_1 + 6776) q^{52} + (1196 \beta_{2} + 360 \beta_1 + 7126) q^{53} + 3025 q^{55} + (546 \beta_{2} + 3100 \beta_1 + 12122) q^{56} + ( - 322 \beta_{2} - 1130 \beta_1 - 10284) q^{58} + (2864 \beta_{2} + 4928 \beta_1 + 1988) q^{59} + ( - 616 \beta_{2} + 432 \beta_1 + 8262) q^{61} + ( - 12 \beta_{2} - 5352 \beta_1 + 4356) q^{62} + (2373 \beta_{2} + 1019 \beta_1 + 10694) q^{64} + ( - 1100 \beta_{2} + 600 \beta_1 - 2250) q^{65} + ( - 64 \beta_{2} + 3360 \beta_1 + 4524) q^{67} + ( - 2108 \beta_{2} - 218 \beta_1 + 24538) q^{68} + (350 \beta_{2} + 1700 \beta_1 + 6950) q^{70} + ( - 3576 \beta_{2} + 3376 \beta_1 - 1552) q^{71} + (1028 \beta_{2} - 1656 \beta_1 + 846) q^{73} + ( - 1096 \beta_{2} + 3702 \beta_1 - 10670) q^{74} + ( - 154 \beta_{2} + 744 \beta_1 + 1842) q^{76} + (726 \beta_{2} + 2420 \beta_1 + 1694) q^{77} + ( - 4202 \beta_{2} - 8252 \beta_1 - 8130) q^{79} + (1675 \beta_{2} - 875 \beta_1 + 4650) q^{80} + (1474 \beta_{2} + 5234 \beta_1 + 25764) q^{82} + ( - 2800 \beta_{2} + 5744 \beta_1 + 15284) q^{83} + ( - 2550 \beta_{2} + 5900 \beta_1 + 2500) q^{85} + (722 \beta_{2} - 6288 \beta_1 + 29686) q^{86} + (242 \beta_{2} + 4477 \beta_1 - 2541) q^{88} + ( - 2640 \beta_{2} - 8896 \beta_1 + 66838) q^{89} + (1436 \beta_{2} - 5496 \beta_1 - 29716) q^{91} + (2560 \beta_{2} + 4752 \beta_1 - 112) q^{92} + (1908 \beta_{2} + 4544 \beta_1 + 21340) q^{94} + (1150 \beta_{2} - 4300 \beta_1 + 24850) q^{95} + ( - 48 \beta_{2} - 13696 \beta_1 + 87090) q^{97} + (2552 \beta_{2} - 3415 \beta_1 + 51839) q^{98}+O(q^{100})$$ q + (b1 - 1) * q^2 + (b2 - b1 - 14) * q^4 - 25 * q^5 + (-6*b2 - 20*b1 - 14) * q^7 + (-2*b2 - 37*b1 + 21) * q^8 + (-25*b1 + 25) * q^10 - 121 * q^11 + (44*b2 - 24*b1 + 90) * q^13 + (-14*b2 - 68*b1 - 278) * q^14 + (-67*b2 + 35*b1 - 186) * q^16 + (102*b2 - 236*b1 - 100) * q^17 + (-46*b2 + 172*b1 - 994) * q^19 + (-25*b2 + 25*b1 + 350) * q^20 + (-121*b1 + 121) * q^22 + (-64*b2 - 688*b1 + 464) * q^23 + 625 * q^25 + (-68*b2 + 486*b1 - 850) * q^26 + (138*b2 + 236*b1 - 318) * q^28 + (-330*b2 - 652*b1 + 1840) * q^29 + (-44*b2 - 56*b1 - 4956) * q^31 + (166*b2 + 395*b1 + 645) * q^32 + (-338*b2 + 818*b1 - 4728) * q^34 + (150*b2 + 500*b1 + 350) * q^35 + (648*b2 - 448*b1 - 2130) * q^37 + (218*b2 - 1408*b1 + 4286) * q^38 + (50*b2 + 925*b1 - 525) * q^40 + (330*b2 + 1804*b1 + 2264) * q^41 + (618*b2 + 1340*b1 - 11850) * q^43 + (-121*b2 + 121*b1 + 1694) * q^44 + (-624*b2 - 112*b1 - 11648) * q^46 + (-364*b2 + 1544*b1 + 7820) * q^47 + (280*b2 + 2832*b1 - 5935) * q^49 + (625*b1 - 625) * q^50 + (-854*b2 - 694*b1 + 6776) * q^52 + (1196*b2 + 360*b1 + 7126) * q^53 + 3025 * q^55 + (546*b2 + 3100*b1 + 12122) * q^56 + (-322*b2 - 1130*b1 - 10284) * q^58 + (2864*b2 + 4928*b1 + 1988) * q^59 + (-616*b2 + 432*b1 + 8262) * q^61 + (-12*b2 - 5352*b1 + 4356) * q^62 + (2373*b2 + 1019*b1 + 10694) * q^64 + (-1100*b2 + 600*b1 - 2250) * q^65 + (-64*b2 + 3360*b1 + 4524) * q^67 + (-2108*b2 - 218*b1 + 24538) * q^68 + (350*b2 + 1700*b1 + 6950) * q^70 + (-3576*b2 + 3376*b1 - 1552) * q^71 + (1028*b2 - 1656*b1 + 846) * q^73 + (-1096*b2 + 3702*b1 - 10670) * q^74 + (-154*b2 + 744*b1 + 1842) * q^76 + (726*b2 + 2420*b1 + 1694) * q^77 + (-4202*b2 - 8252*b1 - 8130) * q^79 + (1675*b2 - 875*b1 + 4650) * q^80 + (1474*b2 + 5234*b1 + 25764) * q^82 + (-2800*b2 + 5744*b1 + 15284) * q^83 + (-2550*b2 + 5900*b1 + 2500) * q^85 + (722*b2 - 6288*b1 + 29686) * q^86 + (242*b2 + 4477*b1 - 2541) * q^88 + (-2640*b2 - 8896*b1 + 66838) * q^89 + (1436*b2 - 5496*b1 - 29716) * q^91 + (2560*b2 + 4752*b1 - 112) * q^92 + (1908*b2 + 4544*b1 + 21340) * q^94 + (1150*b2 - 4300*b1 + 24850) * q^95 + (-48*b2 - 13696*b1 + 87090) * q^97 + (2552*b2 - 3415*b1 + 51839) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{2} - 42 q^{4} - 75 q^{5} - 68 q^{7} + 24 q^{8}+O(q^{10})$$ 3 * q - 2 * q^2 - 42 * q^4 - 75 * q^5 - 68 * q^7 + 24 * q^8 $$3 q - 2 q^{2} - 42 q^{4} - 75 q^{5} - 68 q^{7} + 24 q^{8} + 50 q^{10} - 363 q^{11} + 290 q^{13} - 916 q^{14} - 590 q^{16} - 434 q^{17} - 2856 q^{19} + 1050 q^{20} + 242 q^{22} + 640 q^{23} + 1875 q^{25} - 2132 q^{26} - 580 q^{28} + 4538 q^{29} - 14968 q^{31} + 2496 q^{32} - 13704 q^{34} + 1700 q^{35} - 6190 q^{37} + 11668 q^{38} - 600 q^{40} + 8926 q^{41} - 33592 q^{43} + 5082 q^{44} - 35680 q^{46} + 24640 q^{47} - 14693 q^{49} - 1250 q^{50} + 18780 q^{52} + 22934 q^{53} + 9075 q^{55} + 40012 q^{56} - 32304 q^{58} + 13756 q^{59} + 24602 q^{61} + 7704 q^{62} + 35474 q^{64} - 7250 q^{65} + 16868 q^{67} + 71288 q^{68} + 22900 q^{70} - 4856 q^{71} + 1910 q^{73} - 29404 q^{74} + 6116 q^{76} + 8228 q^{77} - 36844 q^{79} + 14750 q^{80} + 84000 q^{82} + 48796 q^{83} + 10850 q^{85} + 83492 q^{86} - 2904 q^{88} + 188978 q^{89} - 93208 q^{91} + 6976 q^{92} + 70472 q^{94} + 71400 q^{95} + 247526 q^{97} + 154654 q^{98}+O(q^{100})$$ 3 * q - 2 * q^2 - 42 * q^4 - 75 * q^5 - 68 * q^7 + 24 * q^8 + 50 * q^10 - 363 * q^11 + 290 * q^13 - 916 * q^14 - 590 * q^16 - 434 * q^17 - 2856 * q^19 + 1050 * q^20 + 242 * q^22 + 640 * q^23 + 1875 * q^25 - 2132 * q^26 - 580 * q^28 + 4538 * q^29 - 14968 * q^31 + 2496 * q^32 - 13704 * q^34 + 1700 * q^35 - 6190 * q^37 + 11668 * q^38 - 600 * q^40 + 8926 * q^41 - 33592 * q^43 + 5082 * q^44 - 35680 * q^46 + 24640 * q^47 - 14693 * q^49 - 1250 * q^50 + 18780 * q^52 + 22934 * q^53 + 9075 * q^55 + 40012 * q^56 - 32304 * q^58 + 13756 * q^59 + 24602 * q^61 + 7704 * q^62 + 35474 * q^64 - 7250 * q^65 + 16868 * q^67 + 71288 * q^68 + 22900 * q^70 - 4856 * q^71 + 1910 * q^73 - 29404 * q^74 + 6116 * q^76 + 8228 * q^77 - 36844 * q^79 + 14750 * q^80 + 84000 * q^82 + 48796 * q^83 + 10850 * q^85 + 83492 * q^86 - 2904 * q^88 + 188978 * q^89 - 93208 * q^91 + 6976 * q^92 + 70472 * q^94 + 71400 * q^95 + 247526 * q^97 + 154654 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 26x + 8$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 17$$ v^2 - v - 17
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 17$$ b2 + b1 + 17

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −4.78415 0.305203 5.47894
−5.78415 0 1.45634 −25.0000 0 17.6498 176.669 0 144.604
1.2 −0.694797 0 −31.5173 −25.0000 0 83.1683 44.1316 0 17.3699
1.3 4.47894 0 −11.9391 −25.0000 0 −168.818 −196.801 0 −111.974
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$+1$$
$$11$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.6.a.b 3
3.b odd 2 1 165.6.a.c 3
15.d odd 2 1 825.6.a.g 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.a.c 3 3.b odd 2 1
495.6.a.b 3 1.a even 1 1 trivial
825.6.a.g 3 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + 2T_{2}^{2} - 25T_{2} - 18$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(495))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 2 T^{2} + \cdots - 18$$
$3$ $$T^{3}$$
$5$ $$(T + 25)^{3}$$
$7$ $$T^{3} + 68 T^{2} + \cdots + 247808$$
$11$ $$(T + 121)^{3}$$
$13$ $$T^{3} - 290 T^{2} + \cdots + 132063592$$
$17$ $$T^{3} + \cdots - 2547052488$$
$19$ $$T^{3} + 2856 T^{2} + \cdots + 137703680$$
$23$ $$T^{3} + \cdots + 15777349632$$
$29$ $$T^{3} + \cdots + 44413548456$$
$31$ $$T^{3} + \cdots + 121645522944$$
$37$ $$T^{3} + \cdots + 28013661736$$
$41$ $$T^{3} + \cdots - 119305168392$$
$43$ $$T^{3} + \cdots - 38997547520$$
$47$ $$T^{3} + \cdots + 679997104128$$
$53$ $$T^{3} + \cdots + 4393759072056$$
$59$ $$T^{3} + \cdots + 20798004639936$$
$61$ $$T^{3} + \cdots + 43064794504$$
$67$ $$T^{3} + \cdots + 1826752720192$$
$71$ $$T^{3} + \cdots - 34155066048000$$
$73$ $$T^{3} + \cdots - 163103734088$$
$79$ $$T^{3} + \cdots - 70772253539328$$
$83$ $$T^{3} + \cdots + 70386077185728$$
$89$ $$T^{3} + \cdots + 16088649675432$$
$97$ $$T^{3} + \cdots - 148869121092488$$